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3. Poisson process has intensity of a > 0 arrivals/occurrences per hour. Suppose that exactly one arrival/occurrence
occurs in the first hour. Let T be the time at which this arrival/occurrence happens, measured in hours (Thus,
the possible values of T are real numbers in the interval 0, 1)
(a) What is P (0 ST<t) for t e [0, 1)? Note that we are conditioning on exactly one occurrence/arrival during
the first hour
Hint: Consider the mumber of occurrences/arrivals in the interval [0, t and in the interval (t, 1, and then it
becomes similar to Q2 above.
(b) What is the cumulative distribution then of T under the same conditions?
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Transcribed Image Text

3. Poisson process has intensity of a > 0 arrivals/occurrences per hour. Suppose that exactly one arrival/occurrence occurs in the first hour. Let T be the time at which this arrival/occurrence happens, measured in hours (Thus, the possible values of T are real numbers in the interval 0, 1) (a) What is P (0 ST<t) for t e [0, 1)? Note that we are conditioning on exactly one occurrence/arrival during the first hour Hint: Consider the mumber of occurrences/arrivals in the interval [0, t and in the interval (t, 1, and then it becomes similar to Q2 above. (b) What is the cumulative distribution then of T under the same conditions?